Fig. REF  shows the comparison of the spectrum obtained from \(H_{eff}^{\mathbf {K}}\) , truncating to a different number of \(q\)  shells in the inter-layer tunneling terms. For the rigid twist (i.e. when the lattice relaxation is absent), the approximation of including only the innermost \(q\)  shell gives the spectrum that is almost identical with the one produced by the tight binding model in most of the moire BZ, with the mismatch of only \(\sim 2\) meV around the center of the moire Brillouin zone point \(\Gamma \) ; the bandwidth of the narrow bands for the rigid twist is about 40meV and 20meV for the models in Ref. [1]} and Ref. [2]} respectively, with at least one of the bandgaps to the remote bands vanishing. All these features have been well reproduced by including only one \(q\)  shell in \(H^\mathbf {K}_{eff}\) . To further improve the agreement, we include the first two \(q\)  shells and achieve the accuracy presented in Fig. REF  and REF . The excellent agreement obtained with only 2 shells reflects the fact that the Fourier transform of the inter-layer hopping quickly decays as the function of the momentum [3]}, [4]}.
